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\title{First-Year Project Outline}
\author{Alameddin, Anwar}%\\ \\Supervisor: Dr Guletskii, Vladimir}

%%\email{anwar.alameddin@liv.ac.uk}
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\section{Introduction}
We initially want to construct the usual motivic measure of counting point of varieties over a finite field in $\AF^1$-Homotopy theory, then to study the possibility of constructing other motivic measures using the same method. Having the aim of understanding the construction of $\AF^1$-Homotopy category, I studied the category of abelian sheaves on Grothendieck topology, and \'etale topology. Then, I am moving towards the construction of \'etale cohomology, and my next aim is to study Nisnevich, model structure, and localization. I elaborated a detailed study of assertions presented hereafter, providing examples and counter examples, where applicable. I developed full proofs of some assertions, proved briefly in the literature, most of which is already typed.

\section{Grothendieck topology}
\noindent Grothendieck topology generalise the notion topology to any category with all pullbacks, by identifying the functorial properties satisfied by open covers of topological spaces. That in turn facilitate defining sheaves on such categories, with respect to this topology, which allows developing cohomology theories on these categories, given the sheaves considered are abelian. In this section I will be mainly following \cite{Lev07}, \cite{Tam94} and \cite{GelMan03}.
\begin{definition}[Grothendieck Pre-topology]
\noindent Let $\bcC$ be a category, $\tau$ a class of collections of morphisms of $(\bcC\downarrow X), X\in \bcC$. We say that $\tau$ is a Grothendieck pre-topology on $\bcC$ if these collections include all isomorphisms, and are stable under base change and composition. We call each of these collections a covering.
\end{definition}
\noindent A Grothendieck pre-topology is called Grothendieck topology if it is closed under precomposition. 
One notice that Grothendieck pre-topology plays the role of basis of topology in topological spaces. That, every Grothendieck pre-topology defines uniquely a minimum Grothendieck topology that contains it. From now on, topology will refer to Grothendieck topology, and a category $\bcC$ with topology $\tau$ will be called a site, and denoted $\bcC_{\tau}$.\\

\noindent Given two sites, one can define continuous functor between them, to be covering preserving functors that respect base change.\\

\noindent The category of open covering of open subsets in topological spaces forms a pre-order category, such that every finite collection of objects in this category has a refinement. Similar construction is provided for covering of $U\in \bcC_{\tau}$ as follows:

\begin{definition}[Refinement Morphisms]
\noindent Given two coverings $\bcU=\{f_i:U_i\rightarrow U\}_{i\in I}, \bcV=\{g_j:V_j\rightarrow U\}_{j\in J}$. We define a refinement morphism $\phi_{\epsilon}:\bcV\rightarrow \bcU$ to be a pair $\phi_{\epsilon}=(\epsilon,\{\phi_j)\}_{j\in J}$), where $\epsilon:J\rightarrow I$ is a map of sets, and $\phi_j:V_j\rightarrow U_{\epsilon(j)}$ is a $U$-morphism in $\bcC$, $\forall j\in J$.
\end{definition}
\noindent We denote the category of coverings of $U$ and their refinement morphisms by $\tau_U$. This category will be of great use in the construction of \v Cech cohomology, and hence the sheafification as it will follow.
\subsection{Sheaves on sites}
\noindent An $\bcA$-valued pre-sheaves on a site $\bcC_{\tau}$, are defined to be functor $\bcP:\bcC^{op}\rightarrow\bcA$. As an analogy of the case (pre-)sheaves on topological spaces, we call elements of $\bcP(X)$ sections, whenever the objects of $\bcA$ are set-theoretical objects. The sheaves on topological spaces was defined to be pre-sheaves that allows unique gluing of sections. Examining the functorial properties of unique gluing of sections, $\bcA$-valued sheaves on $\bcC$ was defined, for any category $\bcA$ with all products, as follows:
\begin{definition}
\noindent Let $\bcS:\bcC_{\tau}^{op}\rightarrow \bcA$ be a pre-sheaf, where $\bcA$ has all products. We say that $\bcS$ is a sheaf if for all objects $U\in \bcC$ and all coverings$\{f_i:U_i\rightarrow U\}_{i \in I}$ the following diagram is exact:\\
\begin{equation}\label{Sheaf}
\xymatrix{ \bcS(U)\stackrel{\rho_{\bcU}}{\rightarrow}\displaystyle\prod_{i\in I}\bcS(U_i)\ar@<-3pt>[r]_{\rho_{\bcU,2}}\ar@<3pt>[r]^{\rho_{\bcU,1}} &\displaystyle\prod_{i,j\in I}\bcS(U_i\times_U U_j)}
\end{equation}
\noindent Where $\rho_{\bcU}, \rho_{\bcU,1}, \text{ and }\rho_{\bcU,2}$ are the canonical morphisms induced by the universal property of product in $\bcA$, and the base change stability of coverings in $\tau$ .
\end{definition}
\noindent The diagram \eqref{Sheaf} is defined for any pre-sheaf, and is commutative, in the above settings. Notice that, the exactness of \eqref{Sheaf} implies that $\rho_{\bcU}$ is a monomorphism, if $\bcA$ is abelian. Such pre-sheaves with $\rho_{\bcU}$ is a monomorphism for every covering, are called separated pre-sheaves, which arise naturally in the sheafification of abelian pre-sheaves.

\begin{example}[The canonical topology]
\noindent The collections of effective epimorphisms in a category $\bcC$ forms a topology, called the canonical topology on $\bcC$. It is the {largest} topology that makes all Yoneda embedding on $\bcC$ into sheaves.
\end{example}

\noindent Our main interest is abelian sheaves (that takes vales in abelian categories, like $\mathbf{AB},R-\mathbf{Mod},...$), to which I will restrict the study in the rest of this section. From now on, $\bcC_{\tau}$ will be a site, and $\bcA$ an abelian category.\\

\noindent Let $\mathbf{Pre}^{\bcA}(\bcC), \mathbf{Shv}_{\tau}^{\bcA}(\bcC)$ denote the categories of $\bcA$-valued pre-sheaves, sheaves, respectively. Where, the morphisms of these categories are the natural transformations of functors. Then, we have the faithfully full inclusion
$$
\goi: \mathbf{Shv}_{\tau}^{\bcA}(\bcC)\rightarrow \mathbf{Pre}^{\bcA}(\bcC)
$$
\noindent This functor admits a left adjoint, which associates an abelian sheaf to each pre-sheaf in natural way through the procedure of sheafification, as follows:\\
\noindent Given an abelian pre-sheaf $\bcP:\bcC^{op}_{\tau}\rightarrow\bcA$, $U\in \bcC$ we define the functor
$$
\bcP_U:\tau^{op}_U\rightarrow \bcA
$$
\noindent Given on the objects by:
$$
\bcP_U(\bcU)=\Ker(\xymatrix{ \displaystyle\prod_{i\in I}\bcP(U_i)\ar@<-3pt>[rr]_{\rho_{\bcU,2}}\ar@<3pt>[rr]^{\rho_{\bcU,1}} &&\displaystyle\prod_{i,j\in I}\bcP(U_i\times_U U_j)})
$$
\noindent Where $\bcU=\{f_i:U_i\rightarrow U\}_{i\in I}$, and defined on morphism using the universal property of product, and kernel in $\bcA$.\\

\noindent One of the important property of this functor is that it identifies different refinement morphisms between two given coverings of an object $U$, which allows us to work with the associated pre-ordered category to $\tau_U$, as of the case of topological spaces. That for any refinement morphisms $\phi_{\epsilon},\psi_{\delta}:\bcU'\rightarrow \bcU$, we have $\bcP_U(\phi_{\epsilon})=\bcP_U(\psi_{\delta})$. This identification allows us to define the pre-sheaf:\\
$$
\bcP^{+}:\bcC_{\tau}^{op}\rightarrow \bcA
$$
\noindent Given on objects by $$\bcP^{+}(U)=\displaystyle\Colim_{\tau_U}\bcP_U$$
And is defined on morphisms using universal properties. Hence, we have the functor $-^{+}:\mathbf{Pre}^{\bcA}(\bcC)\rightarrow \mathbf{Pre}^{\bcA}(\bcC)$, that associate $\bcP^{+}$ to the pre-sheaf $\bcP$, defined in the natural way.\\

\noindent The above construction is an analogue of the sheafification of pre-sheaves on topological spaces. In contrast to the case of topological spaces, where sheafification once gives the associated sheaf, in the case of general Grothendieck topology the above construction does not guarantee that $\bcP^{+}$ is a sheaf. However, the following proposition uses the above construction twice to give the desired sheaf.\\
\begin{proposition}[Sheafification] Let $\bcP\in\mathbf{Pre}^{\bcA}(\bcC)$, be a pre-sheaf, then:
\begin{itemize}
\item The pre-sheaf $\bcP^{+}$ is a separated pre-sheaf.
\item If $\bcP$ is separated pre-sheaf, then the pre-sheaf $\bcP^{+}$ is a sheaf.
\item The functor $-^{+}:\mathbf{Pre}^{\bcA}(\bcC)\rightarrow \mathbf{Pre}^{\bcA}(\bcC)$ is left exact.
\end{itemize}
\end{proposition}

\noindent Since \eqref{Sheaf} is commutative for any abelian pre-sheaf, then $\exists!\eta_{\bcP,\bcU}:\bcP(U)\rightarrow \bcP_U(\bcU)$ a monomorphism that factors $\rho_{\bcU}$. Hence, $\exists!\eta_{\bcP,U}:\bcP(U)\rightarrow \bcP^{+}(U)$ a monomorphism such that $\eta_{\bcP,U}=i_{\bcP,\bcU}\eta_{\bcP,\bcU}$, for every covering $\bcU$ of $U$, where $i_{\bcP,\bcU}$ are the colimit canonical injections. The uniqueness of such $\eta_{\bcP,U}$, defines a monomorphism of pre-sheaves $\eta_{\bcP}:\bcP\rightarrow \bcP^{+}$. This morphism provides the universal property needed to construct the desired left adjoint:

\begin{proposition}
Let $\bcS$ be an abelian sheaf, $\phi:\bcP\rightarrow\bcS$ a morphism of per-sheaves, then:
\begin{itemize}
\item $\phi$ factors uniquely through $\bcP^{+}$.
\item $\phi$ factors uniquely through $\bcP^{\#}$.
\end{itemize}
\end{proposition}

\noindent The above universal property implies the existence of $-^{\#}:\mathbf{Pre}^{\bcA}(\bcC)\rightarrow\mathbf{Shv}_{\tau}^{\bcA}(\bcC)$ a left adjoint of $\goi$, that associate the sheaf $\bcP^{\#}$ to the pre-sheaf $\bcP$.\\

\noindent The category of $\bcA$-valued functors is an abelian category, where the zero, (co)kernel, (co)product functors are defined object-wise. Hence, the category of abelian pre-sheaves is also abelian. In order to study the properties of the category of abelian sheaves, one observes that $\mathbf{Shv}_{\tau}^{\bcA}(\bcC)$  is a full subcategory of $\mathbf{Pre}^{\bcA}(\bcC)$, and that the pre-sheaf kernel of a morphism of abelian sheaves also satisfies the definition the sheaf kernel. Although the pre-sheaf cokernel of morphism of abelian sheaves is not necessary a sheaf, its sheafification satisfies the definition of the sheaf cokernel. Then, one can readily prove the following theorem:

\begin{theorem} Let $\bcC_{\tau}$ be a site, $\bcA$ an abelian category:
\begin{itemize}
\item The category of abelian sheaves  $\mathbf{Shv}_{\tau}^{\bcA}(\bcC)$ is an abelian category.
\item The above right adjoint functor $\goi$ is left exact, and the left adjoint functor $-^{\#}$ is exact.
\end{itemize}
\end{theorem}
\section{\'Etale topology}
\noindent After examining sheaves on Grothendieck topology in general, I moved toward studying \'etale cohomology, which motivates the study of flatness and ramification in order to gain understanding of \'etale topology.
\subsection{Scheme-theoretical closure}
\noindent Given a morphism of schemes $f:X\rightarrow Y$, the set $Im f$ is not necessary open subsets of $Y$, therefore, it doest necessary have a sheaf of structure that makes it into a scheme that can be induced canonically from $\bcO_X,\bcO_Y$, and $f$.\\

\noindent Let $\mathbf{Sch}(Y)$, be the category of schemes over a scheme $Y$, $f:X\rightarrow Y$ a morphism of schemes, and let $\mathbf{ClSch}_f(Y)$ the full subcategory of $(f\uparrow \mathbf{Sch}(Y))$, with objects being closed sub-schemes of $Y$. Then, the scheme-theoretical image of $f$ is defined to be the initial object of $\mathbf{ClSch}_f(Y)$. Given an open immersion $i:X\hookrightarrow Y$, the scheme-theoretical closure of $X$ in $Y$ is defined to be scheme-theoretical image of $i$.
%\subsection{Embedded Components}
%\noindent \tcr{Sothing to come here}.
\subsection{Flatness}
\noindent The notion of flatness provide an analogue of continuous maps between topological spaces, and makes sense of continuously varying family of schemes. In this section I will mainly follow \cite{Eisenbud00} and \cite{Har77}, and used \cite{Eisenbud95} and \cite{Atiyah69}.\\

\noindent Family of schemes is defined to be a morphism of scheme $\pi:X\rightarrow B$, where the fibers of this morphism is though of to form the family, parametrised by the scheme $B$. When the base $B$ is a non-singular one dimensional scheme, the continuity of the family is defined is an analogously to the continuity of maps between topological spaces, as follows:\\

\noindent We define a family of closed sub-schemes of a given scheme $A$, parametrised by $B$ to be a family $$\pi:X\stackrel{i}{\nhookrightarrow} A\times B\stackrel{\pi_B}{\rightarrow} B$$
\noindent Where $i$ is a closed immersion. For a closed point $0\in B$, We have the closed family
$$\pi^{\ast}:X^{\ast}\nhookrightarrow A\times B^{\ast}\stackrel{\pi_{B^{\ast}}}{\rightarrow} B^{\ast}$$
\noindent Where, $B^{\ast}\hookrightarrow B$ is the punctured scheme at $0$, $X^{\ast}$ its preimage. Let $\overline{X^{\ast}}$ be the image of the morphism $X^{\ast}\nhookrightarrow A\times B^{\ast} \hookrightarrow A\times B$, i.e. the scheme-theoretical closure. Hence, we have the closed family
$$\pi':\overline{X^{\ast}}\nhookrightarrow A\times B\stackrel{\pi_B}{\rightarrow} B$$
\noindent Then we define the limit of the family $\pi$ at $0$ to be:
$$
\displaystyle\lim_{b\rightarrow 0}\pi^{-1}(b):=\pi'^{-1}(0)
$$
\noindent And we say that $\pi$ is continuous at $0$ if $\displaystyle\lim_{b\rightarrow 0}\pi^{-1}(b)=\pi^{-1}(0)$.\\

\noindent Serre introduced the notion of flatness to extend the notion of continuous family of sub-schemes into more general settings:
\begin{definition}[Flat family]
\noindent Let $\pi:X\rightarrow B$ be a scheme morphism, we say that $\pi$ flat at $x\in X$ if $\bcO_{X,x}$ is flat over $\bcO_{B,\pi(x)}$, where $\bcO_{X,x}$ is thought of an $\bcO_{B,\pi(x)}$-module through the ring homomorphism:
$$\pi^{\#}_{x}: \bcO_{X,x}\rightarrow\bcO_{B,\pi(x)}$$
\noindent We say that $\pi$ is flat if it is flat at all points $x\in X$.
\end{definition}

\noindent Considering the same settings where continuity is defined, the notion of continuity and flat morphisms coincide, as the following proposition shows:
\begin{proposition}
\noindent Let $A, B$ be schemes, $B$ non-singular, one dimensional, $\pi:X\stackrel{i}{\nhookrightarrow} A\times B\stackrel{\pi_B}{\rightarrow} B$ a closed family, $0$ closed in $B$, $x_0\in X$, such that $\pi(x_0)=0$. Then, the following statements are equivalent:
\begin{itemize}
\item $\pi$ is flat at $x_0$.
\item $\pi$ is continuous at $0$.
\item $\overline{X^{\ast}}$ does not have irreducible or embedded component which is supported on $\pi^{-1}(0)$.
\end{itemize}
\end{proposition}
\noindent The collections of flat morphism of schemes over a scheme $S$ forms a Grothendieck pre-topology on the category of schemes over $S$, $\mathbf{Sch}(S)$. 
\subsection{Ramification}
\subsubsection{Ramifications of holomorphic maps on Riemann surfaces}
\noindent I started with analytical study of ramification, where I followed \cite{Mir95}\\

\noindent Every non-constant holomorphic map between Riemann surfaces $f:X\rightarrow Y$, $p\in X$, $f$ can be expresses locally uniquely by a monomial of degree $m\geq 1$, in a coordinate systems centred around $p$, and $f(p)$ in $X$, and $Y$, respectively. Such $m$ is called the multiplicity of $f$ at $p$ and denoted by $mult_p(f)$. We say that $p$ is a ramification point for $f$ if $mult_p(f)>1$. That, in these coordinate systems we notice that there is a neighbourhood $V$ centred at $f(p)$ with $m$ pre-images for all points of $V\backslash\{f(p)\}$, and a unique pre-image for $f(p)$ with multiplicity $m$. $V\backslash\{f(p)\}$ is holomorphically isomorphic to the $m^{th}$-components of $f^{-1}(V\backslash\{f(p)\})$, which means that $f$ fails locally to form an $m$ covering because of its behaviour at $p$.\\

\noindent Considering compact Riemann surfaces, holomorphic maps behave more nicely in the sense if pre-images are counted with multiplicity that would be an invariant of the map, i.e. independent of the choice of the point for which pre-images are counted, such invariant is called the degree of the map. Moreover, such mops has finitely many Ramification point, which makes the following formula well-defined:


\begin{theorem} [Hurwitz's Formula]
\noindent Let $f:X\rightarrow Y$ be a non-constant holomorphic map between compact Riemann surfaces. Then,
$$2g(X)-2=deg(f)(2g(Y)-2)+\displaystyle\sum_{p\in X}(mult_p(f)-1).$$
\noindent Where $g(x)$, $g(Y)$  are the geniuses of $X$ and $Y$, respectively, and $deg(f)$ is the degree of $f$.
\end{theorem}
\subsubsection{Ramifications of finite morphisms of curves}
\noindent As an analogy of the analytical case, unramified morphisms extends the concept of covering maps on the underlying topological spaces. In this section I mainly follow \cite{Har77} .\\

\noindent Given a finite morphism of non-singular curves $f:X\rightarrow Y$, the stalks $\bcO_{X,x}, \bcO_{Y,f(x)}$ are discrete valuation rings, $\forall x\in X$, having $t$ the local parameter of $\bcO_{X,x}$, and $t'$ the local parameter of $\bcO_{Y,f(x)}$, we have $f^{\#}(t')=u t^m$, for some $m\geq 1$, $u\in \bcO^{\times}_{X,x}$, an analogy of the local normal form in the analytical case. Hence, $m=v_{x}(f^{\#}(t'))$, where $v_x$ is the discrete valuation in $\bcO_{X,x}$, we call such $m$ the multiplicity of $f$ at $x$, and denoted by $mult_x(f)$. We say that $x$ is a ramification at $x$ if $mult_x(f)>1$, otherwise we say that it unramified at $x$. We say that $f$ is unramified iff it is unramified at all points $x\in X$.
\section{Brief of other background study}
\noindent In addition to the above main path, I studied properties of schemes presented in \cite{Har77} and \cite{Eisenbud00}, and some properties of adjunction and their relations with limits and colimits. \\

\noindent In order to achieve my desired goal I need to gain an understanding of different aspect of algebraic geometry. Therefore, I have studied some facts, examples and counter examples in the following subjects:\\



\noindent The canonical sheaf of a smooth variety $X$, of dimension $n$ over a field $k$, is defined to be the $n^{th}$-exterior power of the sheaf of relative differential:
$$
\omega_X=\bigwedge^{n}\Omega_{X/k}
$$
\noindent Then, the canonical sheaf of the projective space $\PR^n$ is $\bcO_{\PR^n}(-n-1)$, and that for $f$, a homogeneous polynomial of degree $d$, the canonical sheaf of the hyper-surface $X=V(f)\subset \PR^n$ is $\omega_{X}=\bcO_{\PR^n}(d-n-1)$.\\

\noindent Invertible sheaves on an ringed space $(X,\bcO_X)$ is defined to be locally free sheaves of rank one. The requirement of the sheaf to be locally free of rank one, is equivalent to defining the sheaf by local sections of the structural sheaf, satisfying the cocycle condition. This fact is an analogy of the equivalence between defining line bundles using of trivialisations or by means of transition functions. Invertible sheaf generated by $n$ global section on a scheme $X$, are of particular interest that it is the pull-back of $\bcO_{\PR^{n}}(1)$ along some regular morphism $\varphi:X\rightarrow \PR^n$, and that every such morphism makes $\varphi^{\ast}\bcO_{\PR^{n}}(1)$ into an invertible sheaf generated by $n$ global section on a scheme $X$.\\

\noindent In addition, I studied other classical concepts of algebraic geometry, with particular focus on rational mapping, tangent spaces and simple singularity of varieties.\\

\noindent Realising my current goal requires an understanding of the hodge theory. Moving toward that direction, I started with studying real and complex vector bundles of differential manifolds, through two approaches: trivialisations and transition functions. Then, I studied almost complex structures on differential manifolds of an even dimension, and the link between complex, real and complexified  tangent and cotangent bundles.\\


\section{Aim of future work}
My main aim is to study the possibility of constructing different motivic measures using $\AF^1$-homotopy category. Therefore, first, I will continue toward obtaining understating of the construction of the category, where I will be using Mumford, Tamme to study \'etale cohomology, Voevodsky to study Nisnevich topology, Gabriel and Zisman to study simplicial sets, Quillen to study Model Structure, Hirschhorn to study Localization. Afterwards, I will be working on the construction of the motivic measures, and proving that it satisfies the desired properties.	
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